Consider a uniform sphere that has mass and radius . The gravitational energy of an isotropic sphere is given by so using we obtain
The energy is given by and defining we obtain
where . So
Using we can write
Where and we focus on the function , the plot is given by
The function looks like a circle. Note that and – so we consider – the plot is given by
We therefore propose .
The total energy is given by
We assume that and that is constant in time, then we obtain
The solution is given by
Or using and we can write
Then and . So we obtain
A plot of the radius of the universe in time:
And a plot of the speed of the radius of the universe in time:
The radius is accelerating at the beginning. Is it possible that we are just at the beginning of the universe so that the expansion is still accelerating?
Here we have just used special relativity. The total energy of the universe in this model simply is zero (kinetic energy is positive – but the gravitational energy is negative) – so before the ‘big bang’ there was simply nothing – not energy – no matter and no time.